How to have a force field affecting only one particle system hair
Fields only have an effect on objects and particles that reside on the same layer. Therefore, one solution is to have particle system 'alpha' on one layer, 'beta' on a second layer, and the force on the same layer as 'alpha'. 

To achieve this you can use M to move objects to different layers - simply press M while an object is selected and the 'Move to Layer' popup should appear. Just select the layer(s) that the object should reside on (an object can be on more than one layer - hold Shift to allow you to select multiple). To 'View' multiple layers, simply hold Shift and click the layers in the button bar at the bottom of the 3d Editor window.

With one of the objects (Beta) on a different layer and both layers selected for view, you should see no difference in the scene. However, the moved object's particles will no longer be affected by the force.